= Elementary predictable process with stopping-time intervals
{title2=$H=\sum_jh_j\mathbf1_{(\tau_j,\tau_{j+1}]}$}
= Elementary predictable processes with stopping-time intervals
{synonym}
An elementary <predictable process> may use finitely many ordered <stopping times> $\tau_j$, with bounded $\mathcal F_{\tau_j}$-measurable coefficients $h_j$ on $(\tau_j,\tau_{j+1}]$. Its elementary <stochastic integral> is $\sum_jh_j(X_{t\wedge\tau_{j+1}}-X_{t\wedge\tau_j})$. The stopped indicators are left-continuous and adapted, so the process is <predictable>; common refinement shows the sum is independent of its representation. Deterministic endpoints recover the usual <simple predictable process>. These integrands give the precise good-integrator test in the <Bichteler-Dellacherie theorem>.
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